A game in normal form consists of: Set of players = {1, , } - - PowerPoint PPT Presentation

ALGOS TRUTH JUSTICE

Game Theory I: Basic Concepts

Teachers: Ariel Procaccia (this time) and Alex Psomas

NORMAL-FORM GAME

• A game in normal form consists of:
• Set of players 𝑂 = {1, … , 𝑜}
• Strategy set 𝑇
• For each 𝑗 ∈ 𝑂, utility function

𝑣𝑗: 𝑇𝑜 → ℝ: if each j ∈ 𝑂 plays the strategy 𝑡

𝑘 ∈ 𝑇, the utility of player 𝑗

is 𝑣𝑗(𝑡1, … , 𝑡𝑜)

THE PRISONER’S DILEMMA

• Two men are charged with a crime
• They are told that:
• If one rats out and the other does not, the

rat will be freed, other jailed for nine years

• If both rat out, both will be jailed for six

years

• They also know that if neither rats out,

both will be jailed for one year

THE PRISONER’S DILEMMA

• 1,-1
• 9,0

0,-9

• 6,-6

Cooperate Defect Cooperate Defect

What would you do?

ON TV

http://youtu.be/S0qjK3TWZE8

THE PROFESSOR’S DILEMMA

106,106

• 10,0

0,-10 0,0

Make effort Slack off Listen Sleep

Dominant strategies?

Professor Class

NASH EQUILIBRIUM

• In a Nash equilibrium, no player wants

to unilaterally deviate

• Each player’s strategy is a best

response to strategies of others

• Formally, a Nash equilibrium is a vector
• f strategies 𝒕 = 𝑡1 … , 𝑡𝑜 ∈ 𝑇𝑜 such

that for all 𝑗 ∈ 𝑂, 𝑡𝑗

′ ∈ 𝑇,

𝑣𝑗 𝒕 ≥ 𝑣𝑗(𝑡1, … , 𝑡𝑗−1, 𝑡𝑗

′, 𝑡𝑗+1, … , 𝑡𝑜)

THE PROFESSOR’S DILEMMA

106,106

• 10,0

0,-10 0,0

Make effort Slack off Listen Sleep

Nash equilibria?

Professor Class

R P S R

0,0

• 1,1

1,-1

P

1,-1 0,0

• 1,1

S

• 1,1

1,-1 0,0

ROCK-PAPER-SCISSORS

Nash equilibria?

MIXED STRATEGIES

• A mixed strategy is a probability

distribution over (pure) strategies

• The mixed strategy of player 𝑗 ∈ 𝑂 is 𝑦𝑗,

where 𝑦𝑗(𝑡𝑗) = Pr[𝑗 plays 𝑡𝑗]

• The utility of player 𝑗 ∈ 𝑂 is

𝑣𝑗 𝑦1, … , 𝑦𝑜 = ෍

(𝑡1,…,𝑡𝑜)∈𝑇𝑜

𝑣𝑗 𝑡1, … , 𝑡𝑜 ⋅ ෑ

𝑘=1 𝑜

𝑦𝑘(𝑡

𝑘)

EXERCISE: MIXED NE

R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0

• Exercise: player 1 plays

1 2 , 1 2 , 0 , player 2

plays 0,

1 2 , 1 2 . What is 𝑣1?

• Exercise: Both players play

1 3 , 1 3 , 1 3 . What is

𝑣1?

EXERCISE: MIXED NE

Which is a NE?

1.

1 2 , 1 2 , 0 , 1 2 , 1 2 , 0

3.

1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3

2.

1 2 , 1 2 , 0 , 1 2 , 0, 1 2

4.

1 3 , 2 3 , 0

,

2 3 , 0, 1 3

R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0

?

Poll 1

NASH’S THEOREM

• Theorem [Nash, 1950]: In any (finite)

game there exists at least one (possibly mixed) Nash equilibrium

• What about computing a Nash

equilibrium? Stay tuned…

DOES NE MAKE SENSE?

• Two players, strategies are {2, … , 100}
• If both choose the same number, that is

what they get

• If one chooses 𝑡, the other 𝑢, and 𝑡 < 𝑢, the

former player gets 𝑡 + 2, and the latter gets 𝑡 − 2

• Poll 2: What would you choose?

100 99 98 97 96 95

CORRELATED EQUILIBRIUM

• Let 𝑂 = {1,2} for simplicity
• A mediator chooses a pair of strategies

(𝑡1, 𝑡2) according to a distribution 𝑞

• ver 𝑇2
• Reveals 𝑡1 to player 1 and 𝑡2 to player 2
• When player 1 gets 𝑡1 ∈ 𝑇, he knows the

distribution over strategies of 2 is Pr 𝑡2 𝑡1 = Pr 𝑡1 ∧ 𝑡2 Pr 𝑡1 = 𝑞 𝑡1, 𝑡2 Pr[𝑡1]

CORRELATED EQUILIBRIUM

• Player 1 is best responding if for all 𝑡1

′ ∈ 𝑇

෍

𝑡2∈𝑇

Pr 𝑡2 𝑡1 𝑣1 𝑡1, 𝑡2 ≥ ෍

𝑡2∈𝑇

Pr 𝑡2 𝑡1 𝑣1(𝑡1

′, 𝑡2)

• Equivalently,

෍

𝑡2∈𝑇

𝑞 𝑡1, 𝑡2 𝑣1 𝑡1, 𝑡2 ≥ ෍

𝑡2∈𝑇

𝑞 𝑡1, 𝑡2 𝑣1(𝑡1

′, 𝑡2)

• 𝑞 is a correlated equilibrium (CE) if both

players are best responding

• Every Nash equilibrium is a correlated

equilibrium, but not vice versa

GAME OF CHICKEN

http://youtu.be/u7hZ9jKrwvo

GAME OF CHICKEN

Dare Chicken Dare

0,0 4,1

Chicken

1,4 3,3

• Social welfare is the

sum of utilities

• Pure NE: (C,D) and

(D,C), social welfare = 5

• Mixed NE: both

(1/2,1/2), social welfare = 4

• Optimal social welfare

= 6

GAME OF CHICKEN

• Correlated equilibrium:
• (D,D): 0
• (D,C):

1 3

• (C,D):

1 3

• (C,C):

1 3

• Social welfare of CE =

16 3

Dare Chicken Dare

0,0 4,1

Chicken

1,4 3,3

IMPLEMENTATION OF CE

• Instead of a mediator, use a

hat!

• Balls in hat are labeled with

“chicken” or “dare”, each blindfolded player takes a ball

Which balls implement the distribution

• f the previous slide?
• 1. 1 chicken, 1 dare
• 3. 2 chicken, 1 dare
• 2. 1 chicken, 2 dare
• 4. 2 chicken, 2 dare

Poll 3

?

CE AS LP

• Can compute CE via linear

programming in polynomial time!

find ∀𝑡1, 𝑡2 ∈ 𝑇, 𝑞 𝑡1, 𝑡2 s.t .t. ∀𝑡1, 𝑡1

′ ∈ 𝑇,

∀𝑡2, 𝑡2

′ ∈ 𝑇,

෍

𝑡1,𝑡2∈𝑇

𝑞 𝑡1, 𝑡2 = 1 ∀𝑡1, 𝑡2 ∈ 𝑇, 𝑞 𝑡1, 𝑡2 ∈ [0,1]

෍

𝑡2∈𝑇

𝑞 𝑡1, 𝑡2 𝑣1 𝑡1, 𝑡2 ≥ ෍

𝑡2∈𝑇

𝑞 𝑡1, 𝑡2 𝑣1(𝑡1

′, 𝑡2)

෍

𝑡1∈𝑇

𝑞 𝑡1, 𝑡2 𝑣2 𝑡1, 𝑡2 ≥ ෍

𝑡1∈𝑇

𝑞 𝑡1, 𝑡2 𝑣2(𝑡1, 𝑡2

′)